Math Problem Statement
Solve the system of equations: x² + y² = 25 and x² - 4y² = 9.
Solution
To solve the given system of equations for and :
x^2 + y^2 = 25 \quad \text{(1)}\\ x^2 - 4y^2 = 9 \quad \text{(2)} \end{cases}$$ ### Step-by-Step Solution 1. **Subtract Equation (2) from Equation (1):** $$(x^2 + y^2) - (x^2 - 4y^2) = 25 - 9$$ Simplifying: $$x^2 + y^2 - x^2 + 4y^2 = 16$$ This gives: $$5y^2 = 16$$ Therefore: $$y^2 = \frac{16}{5}$$ 2. **Substitute $$y^2 = \frac{16}{5}$$ into Equation (1):** $$x^2 + y^2 = 25$$ Plugging in $$y^2 = \frac{16}{5}$$: $$x^2 + \frac{16}{5} = 25$$ To clear the fraction, multiply everything by 5: $$5x^2 + 16 = 125$$ Solving for $$x^2$$: $$5x^2 = 109$$ Therefore: $$x^2 = \frac{109}{5}$$ 3. **Find the Values of $$x$$ and $$y$$:** - $$x = \pm \sqrt{\frac{109}{5}}$$ - $$y = \pm \sqrt{\frac{16}{5}}$$ Thus, the solutions are: $$(x, y) = \left( \sqrt{\frac{109}{5}}, \sqrt{\frac{16}{5}} \right), \left( \sqrt{\frac{109}{5}}, -\sqrt{\frac{16}{5}} \right), \left( -\sqrt{\frac{109}{5}}, \sqrt{\frac{16}{5}} \right), \left( -\sqrt{\frac{109}{5}}, -\sqrt{\frac{16}{5}} \right)$$ --- ### Final Solutions: $$\left( \sqrt{\frac{109}{5}}, \sqrt{\frac{16}{5}} \right), \quad \left( \sqrt{\frac{109}{5}}, -\sqrt{\frac{16}{5}} \right), \quad \left( -\sqrt{\frac{109}{5}}, \sqrt{\frac{16}{5}} \right), \quad \left( -\sqrt{\frac{109}{5}}, -\sqrt{\frac{16}{5}} \right)$$ Would you like further details on the steps, or have any questions? --- ### Related Questions 1. How would you solve the system if one of the equations were linear? 2. How do you graph the solutions for these types of systems? 3. What does the intersection of these curves represent geometrically? 4. How would this problem change if the constant in the second equation were different? 5. How can substitution methods help in solving nonlinear systems? ### Tip In solving systems of equations, eliminating one variable often simplifies the problem significantly.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Quadratic Equations
Formulas
Equation of a circle: x² + y² = r²
Difference of squares: a² - b² = (a - b)(a + b)
Theorems
Methods for solving systems of equations
Substitution and elimination methods
Suitable Grade Level
Grades 10-12
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